In this chapter, we present a hierarchy of infinite-state systems based on the primitive operations of sequential and parallel composition; the hierarchy includes a variety of commonly-studied classes of systems such as context-free and pushdown automata, and Petri net processes. We then examine the equivalence and regularity checking problems for these classes, with special emphasis on bisimulation equivalence, stressing the structural techniques which have been devised for solving these problems. Finally, we explore the model checking problem over these classes with respect to various linear- and branching-time temporal logics.

Recently there has been a spurt of activity in concurrency theory centred on the analysis of infinite-state systems. The following two problems have been intensely investigated: (1) given two infinite-state systems, are they equal with respect to a certain equivalence notion?, and (2) given an infinite-state system and a property expressed in a certain temporal logic, does the system satisfy the property? In his paper "Infinite Results" [Mol96] , Moller surveys some of the key results on the decidability and complexity of problem (1). This paper is a survey on the results about problem (2). 1 Introduction Most techniques for the verification of concurrent systems proceed by an exhaustive traversal of the state space. Therefore, they are inherently incapable of considering systems with infinitely many states. Recently, some methods have been developed to overcome this limitation, at least for restricted classes of infinite-state systems. Using them, several verification problems have b...

Introduction The model of timed automata, introduced in [1], is obtained from classical finite automata by adding a finite set of real valued variables called clocks. Clock values increase continuously at the same rate as time in the control locations, and they can be tested and reset by transitions. A test consists in comparing clock values, or the difference between two such values, with constants. For these models, the test for emptiness is decidable (and PSpace-complete [2]), which explains their successful use for the verification of timed systems. Extensions have later been proposed in several directions, with the aim to increase the expressive power, while preserving the decidability result. For instance, decidability of emptiness still holds for some classes of timed transition systems (like Petri nets or context-free grammars), where the hypothesis of finite control is removed ([4]). Replacing reset by more general update operations can also preserve decidability ([6

We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series - the semilinear and the bounded series - both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language - the largest set commuting with a language -, of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago - asking whether or not the centralizer of any rational language is rational - in the case of periodic, binary, and ternary sets of words, as well as for rational c-codes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an c-code X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.